Preheat SAUTÉ function on slow cooker.
Beef: In medium bowl, toss beef chuck roast with oil and 1 1/2 tsp adobo all-purpose seasoning with pepper. Transfer to slow cooker and cook, searing each side, for 12 to 18 minutes or until evenly browned all over. Transfer to medium plate.
Add onion, garlic, chipotle peppers in adobo sauce, adobo sauce, ancho chili and guajillo chili to slow cooker. Cook, stirring occasionally, for 8 to 10 minutes. Stir in diced tomatoes, 4 cups water, beef bouillon and remaining 1 1/2 tsp. Adobo.
Transfer beef back to slow cooker, cover and reduce temperature to LOW. Cook, stirring occasionally, for 2 to 3 hours or until beef is very tender and pulls apart easy with a fork.
Gently transfer beef to cutting board. Using 2 forks, shred beef. Transfer to another medium bowl.
Strain broth into another medium bowl (reserving onion-tomato mixture) and stir in 1/4 cup lime juice.
Transfer onion-tomato mixture to blender and add 1/4 cup broth. Pulse until smooth. Add to beef. Add remaining 2 tbsp. lime juice to beef.
In small bowl, toss together onion and cilantro. Transfer 1/4 cup onion mixture to another small bowl and reserve.
Assembly: Heat large skillet over medium heat. Dip a tortilla into strained broth and place in skillet. Top with 1/3 cup quesadilla cheese, 1/4 cup beef mixture, 2 tbsp of remaining onion mixture and fold to enclose filling. Cook, flipping once, for 2 to 4 minutes or until tortilla is crispy and cheese has melted. Repeat with remaining tortillas, more strained broth, remaining quesadilla cheese, remaining beef mixture and more onion mixture.
To medium bowl of broth, add reserved 1/4 cup onion mixture. Stir to combine and transfer to serving bowl.
Arrange quesabirria on serving platter and serve with lime wedges and consommé for dipping.
Note: Any remaining cooking liquid can be used in soups and sauces or for braising pork, beef, or chicken.